I understand the following:

Theorem: Suppose the real functions $\displaystyle u(x,y)$ and $\displaystyle v(x,y)$ are continuous and have continuous 1st order partials in a domain D. If $\displaystyle u$ and $\displaystyle v$ satisfy the CR equations at all points in D, then $\displaystyle f(z) = u(x,y) +iv(x,y)$ is analytic in D.

Therefore, the CR equations $\displaystyle u_{x} = v_{y}$ and $\displaystyle u_{y}=-v_{x}$ must hold.

But I don't understand what I'm supposed to conclude when I do the following problem:

$\displaystyle f(z)=|z-10|^2 = y^2 + (x^2 - 10)^2$

When I do the CR equations I get $\displaystyle v_{y} = 0$ and $\displaystyle v_x=0$. Does that mean it is only differentiable at $\displaystyle (0,0)$? What does that say about being analytic? I've seen a few problems like this were some of the CR equations are equal to zero such as $\displaystyle f(z) = |z|^2$. For that one I understand that limits at different paths are not equal therefore only differentiable at $\displaystyle z = 0$. Do I need to do different paths to show where it's only differentiable for the problem above or how should I try to tackle this problem? Thank you so much!